We study the multiple manifold problem, a binary classification task modeled on applications in machine vision, in which a deep fully-connected neural network is trained to separate two low-dimensional submanifolds of the unit sphere. We provide an analysis of the one-dimensional case, proving for a simple manifold configuration that when the network depth L is large relative to certain geometric and statistical properties of the data, the network width n grows as a sufficiently large polynomial in L, and the number of i.i.d. samples from the manifolds is polynomial in L, randomly-initialized gradient descent rapidly learns to classify the two manifolds perfectly with high probability. Our analysis demonstrates concrete benefits of depth and width in the context of a practically-motivated model problem: the depth acts as a fitting resource, with larger depths corresponding to smoother networks that can more readily separate the class manifolds, and the width acts as a statistical resource, enabling concentration of the randomly-initialized network and its gradients. Along the way, we establish essentially optimal nonasymptotic rates of concentration for the neural tangent kernel of deep fully-connected ReLU networks using martingale techniques, requiring width n \geq L poly(d0) to achieve uniform concentration of the initial kernel over a d0-dimensional submanifold of the unit sphere. Our approach should be of use in establishing similar results for other network architectures.